Fit GAMs to dispersal patterns of Ascochyta conidia

0.1 Import Data

if (!require("pacman")) install.packages("pacman")
pacman::p_load(tidyverse, broom, skimr, devtools, ggpubr, mgcv, extrafont, mgcViz, here)

theme_set(theme_pubclean(base_size = 14))

# load the function to print GAM figures
source(here("R", "p_gam.R"))

# import
lesion <- read_csv(here("cache", "summarised_lesion_data.csv"))
weather <- read_csv(here("cache", "weather_summary.csv"))

dat <- left_join(lesion, weather, by = c("site" = "Location", "rep" = "Rep"))

0.2 Fit GAMs

For reproducibility purposes, use set.seed().

set.seed(27)

0.2.1 mod1 - s(Distance)

mod1 <-
   gam(
      mean_pot_count ~ s(distance, k = 5),
      data = dat
   )

summary(mod1)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.08024    0.04751   22.74   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df    F p-value    
## s(distance) 3.926  3.996 78.4  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.482   Deviance explained = 48.8%
## GCV = 0.76522  Scale est. = 0.75394   n = 334

print(p_gam(x = getViz(mod1)) +
         ggtitle("s(Distance)"),
      pages = 1)

0.2.2 mod2 - s(Distance) + Precipitation

mod2 <-
   gam(
      mean_pot_count ~ sum_rain+ s(distance, k = 5),
      data = dat
   )

summary(mod2)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ sum_rain + s(distance, k = 5)
## 
## Parametric coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.124951   0.055685  20.202   <2e-16 ***
## sum_rain    -0.010853   0.007088  -1.531    0.127    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F p-value    
## s(distance) 3.926  3.996 78.71  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.484   Deviance explained = 49.1%
## GCV = 0.76443  Scale est. = 0.75086   n = 334

print(p_gam(x = getViz(mod2)) +
         ggtitle("s(Distance) + Precipitation"),
      pages = 1)

0.2.3 mod3 - s(Distance) + Windspeed

mod3 <-
   gam(mean_pot_count ~ mws + s(distance, k = 5),
       data = dat)

summary(mod3)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ mws + s(distance, k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.64401    0.11825   5.446 1.01e-07 ***
## mws          0.12273    0.03059   4.012 7.47e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F p-value    
## s(distance) 3.929  3.996 81.99  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.504   Deviance explained = 51.2%
## GCV = 0.73389  Scale est. = 0.72086   n = 334

print(p_gam(x = getViz(mod3)) +
         ggtitle("s(Distance) + Windspeed"),
      pages = 1)

0.2.4 mod4 - s(Distance) + Windspeed + Precipitation

mod4 <-
   gam(mean_pot_count ~ sum_rain + mws + s(distance, k = 5),
       data = dat)

summary(mod4)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ sum_rain + mws + s(distance, k = 5)
## 
## Parametric coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.674729   0.118523   5.693 2.78e-08 ***
## sum_rain    -0.014719   0.006968  -2.112   0.0354 *  
## mws          0.131154   0.030694   4.273 2.53e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##              edf Ref.df     F p-value    
## s(distance) 3.93  3.996 82.84  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.509   Deviance explained = 51.8%
## GCV = 0.72844  Scale est. = 0.71333   n = 334

print(p_gam(x = getViz(mod4)) +
         ggtitle("s(Distance) + Windspeed + Precipitation"),
      pages = 1)

0.2.5 mod5 - s(Distance + Windspeed) + Precipitation

mod5 <-
   gam(
      mean_pot_count ~ sum_rain + s(distance + mws, k = 5),
      data = dat
   )

## Warning in term[i] <- attr(terms(reformulate(term[i])), "term.labels"): number
## of items to replace is not a multiple of replacement length

summary(mod5)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ sum_rain + s(distance + mws, k = 5)
## 
## Parametric coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.124951   0.055685  20.202   <2e-16 ***
## sum_rain    -0.010853   0.007088  -1.531    0.127    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F p-value    
## s(distance) 3.926  3.996 78.71  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.484   Deviance explained = 49.1%
## GCV = 0.76443  Scale est. = 0.75086   n = 334

print(p_gam(x = getViz(mod5)) +
         ggtitle("s(Distance + Windspeed) + Precipitation"),
      pages = 1)

0.2.6 mod6 - s(Distance) + s(Windspeed) + Precipitation

mod6 <-
   gam(
      mean_pot_count ~ sum_rain + s(distance, k = 5) + s(mws, k = 5),
      data = dat
   )

summary(mod6)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ sum_rain + s(distance, k = 5) + s(mws, k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.18789    0.07152  16.608   <2e-16 ***
## sum_rain    -0.02613    0.01379  -1.895    0.059 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F  p-value    
## s(distance) 3.938  3.997 93.81  < 2e-16 ***
## s(mws)      3.932  3.997 15.97 3.05e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.566   Deviance explained = 57.8%
## GCV = 0.64971  Scale est. = 0.63051   n = 334

print(p_gam(x = getViz(mod6)) +
         ggtitle("s(Distance) + s(Windspeed) + Precipitation"),
      pages = 1)

0.2.7 mod7 - s(Distance) + s(Windspeed)

mod7 <-
   gam(
      mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5),
      data = dat
   )

summary(mod7)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.08024    0.04366   24.74   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F  p-value    
## s(distance) 3.937  3.997 92.96  < 2e-16 ***
## s(mws)      3.917  3.995 16.04 6.03e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.562   Deviance explained = 57.3%
## GCV = 0.65392  Scale est. = 0.63659   n = 334

print(p_gam(x = getViz(mod7)) +
         ggtitle("s(Distance) + s(Windspeed)"),
      pages = 1)

0.2.8 mod8 - s(Distance) + s(Windspeed) + s(Precipitation)

mod8 <-
   gam(
      mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5) + s(sum_rain, k = 5),
      data = dat
   )

summary(mod8)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5) + s(sum_rain, 
##     k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.08024    0.04345   24.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df      F p-value    
## s(distance) 3.938  3.997 93.802  <2e-16 ***
## s(mws)      2.977  3.051  6.713  0.0161 *  
## s(sum_rain) 1.931  1.942  1.979  0.1899    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.566   Deviance explained = 57.8%
## GCV = 0.64966  Scale est. = 0.63051   n = 334

print(p_gam(x = getViz(mod8)) +
         ggtitle("s(Distance) + s(Windspeed) + s(Precipitation)"),
      pages = 1)

0.2.9 mod9 - s(Distance) + s(Precipitation)

mod9 <-
   gam(
      mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5),
      data = dat
   )

summary(mod9)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.08024    0.04594   23.51   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F  p-value    
## s(distance) 3.931  3.996 83.82  < 2e-16 ***
## s(sum_rain) 2.030  2.198 10.76 2.31e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.515   Deviance explained = 52.4%
## GCV = 0.71995  Scale est. = 0.70494   n = 334

print(p_gam(x = getViz(mod9)) +
         ggtitle("s(Distance) + s(Precipitation)"),
      pages = 1)

0.2.10 mod10 - s(Distance) +s(Precipitation) + Windspeed

mod10 <-
   gam(
      mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5) + mws,
      data = dat
   )

summary(mod10)

## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5) + mws
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -2.8631     1.4637  -1.956   0.0513 . 
## mws           1.1095     0.4116   2.695   0.0074 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df     F  p-value    
## s(distance) 3.936  3.997 93.81  < 2e-16 ***
## s(sum_rain) 3.819  3.967 11.16 9.66e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.566   Deviance explained = 57.8%
## GCV = 0.64947  Scale est. = 0.6305    n = 334

print(p_gam(x = getViz(mod10)) +
         ggtitle("s(Distance) + s(Precipitation) + Windspeed"),
      pages = 1)

0.2.11 mod11.0 - s(Distance) + s(Windspeed) + s(Precipitation), family = tw()

This is the same as mod8 but using family = tw(), see ?family.mgcv for more on the families. The Tweedie distribution is used where the distribution has a positive mass at zero, but is continuous unlike the Poisson distribution that requires count data. The data visualisation shows clearly that the mean pot count data have this shape.

mod11.0 <-
   gam(
      mean_pot_count ~ s(distance, k = 5) + 
         s(mws, k = 5) + 
         s(sum_rain, k = 5),
      data = dat,
      family = tw()
   )

summary(mod11.0)

## 
## Family: Tweedie(p=1.044) 
## Link function: log 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5) + s(sum_rain, 
##     k = 5)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.22837    0.04099  -5.571 5.32e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df       F  p-value    
## s(distance) 3.497  3.855 123.761  < 2e-16 ***
## s(mws)      2.938  3.054  13.497 1.39e-05 ***
## s(sum_rain) 1.901  1.928   5.573   0.0132 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.673   Deviance explained = 61.2%
## -REML = 309.77  Scale est. = 0.36396   n = 334

print(p_gam(x = getViz(mod11.0)) +
   ggtitle("s(Distance) + s(Windspeed) + s(Precipitation), family = tw()"),
   pages = 1)

0.2.12 mod11.1 - s(Distance, bs = “ts”) + s(Windspeed, bs = “ts”) + s(Precipitation, bs = “ts”), family = tw()

mod11.1 <-
   gam(
      mean_pot_count ~ s(distance, k = 5, bs = "ts") + 
         s(mws, k = 5, bs = "ts") + 
         s(sum_rain, k = 5, bs = "ts"),
      data = dat,
      family = tw()
   )

summary(mod11.1)

## 
## Family: Tweedie(p=1.044) 
## Link function: log 
## 
## Formula:
## mean_pot_count ~ s(distance, k = 5, bs = "ts") + s(mws, k = 5, 
##     bs = "ts") + s(sum_rain, k = 5, bs = "ts")
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.22588    0.04094  -5.518 7.02e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##               edf Ref.df       F  p-value    
## s(distance) 3.250      4 117.921  < 2e-16 ***
## s(mws)      2.875      4  13.403  1.9e-12 ***
## s(sum_rain) 1.839      4   2.995 0.000496 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.671   Deviance explained =   61%
## -REML = 322.33  Scale est. = 0.36412   n = 334

print(
   p_gam(x = getViz(mod11.1)) +
      ggtitle(
         "s(Distance, bs = 'ts') + s(Windspeed, bs = 'ts')\n+ s(Precipitation, bs = 'ts'), family = tw()"
      ),
   pages = 1
)

This model, same structure as mod11.0, uses thin-plate splines to shrink the coefficients of the smooth to zero when possible.

0.3 Compare the Models

0.3.1 AIC, BIC

models <- list(mod1 = mod1,
               mod2 = mod2,
               mod3 = mod3,
               mod4 = mod4,
               mod5 = mod5,
               mod6 = mod6,
               mod7 = mod7,
               mod8 = mod8,
               mod9 = mod9,
               mod10 = mod10,
               mod11.0 = mod11.0,
               mod11.1 = mod11.1
               )
map_df(models, glance, .id = "model") %>%
   arrange(AIC)

## # A tibble: 12 x 7
##    model      df logLik   AIC   BIC deviance df.residual
##    <chr>   <dbl>  <dbl> <dbl> <dbl>    <dbl>       <dbl>
##  1 mod11.0  9.34  -320.  663.  708.     141.        325.
##  2 mod11.1  8.96  -322.  667.  712.     141.        325.
##  3 mod10    9.76  -392.  805.  846.     204.        324.
##  4 mod8     9.85  -392.  805.  847.     204.        324.
##  5 mod6     9.87  -392.  806.  847.     204.        324.
##  6 mod7     8.85  -394.  808.  845.     207.        325.
##  7 mod9     6.96  -412.  840.  870.     231.        327.
##  8 mod4     6.93  -414.  844.  874.     233.        327.
##  9 mod3     5.93  -416.  846.  873.     236.        328.
## 10 mod2     5.93  -423.  860.  886.     246.        328.
## 11 mod5     5.93  -423.  860.  886.     246.        328.
## 12 mod1     4.93  -424.  860.  883.     248.        329.

0.3.2 R²

enframe(c(
   mod1 = summary(mod1)$r.sq,
   mod2 = summary(mod2)$r.sq,
   mod3 = summary(mod3)$r.sq,
   mod4 = summary(mod4)$r.sq,
   mod5 = summary(mod5)$r.sq,
   mod6 = summary(mod6)$r.sq,
   mod7 = summary(mod7)$r.sq,
   mod8 = summary(mod8)$r.sq,
   mod9 = summary(mod9)$r.sq,
   mod10 = summary(mod10)$r.sq,
   mod11.0 = summary(mod11.0)$r.sq,
   mod11.1 = summary(mod11.1)$r.sq
)) %>%
   arrange(desc(value))

## # A tibble: 12 x 2
##    name    value
##    <chr>   <dbl>
##  1 mod11.0 0.673
##  2 mod11.1 0.671
##  3 mod10   0.566
##  4 mod6    0.566
##  5 mod8    0.566
##  6 mod7    0.562
##  7 mod9    0.515
##  8 mod4    0.509
##  9 mod3    0.504
## 10 mod2    0.484
## 11 mod5    0.484
## 12 mod1    0.482

0.3.3 ANOVA

anova(mod1,
      mod2,
      mod3,
      mod4,
      mod5,
      mod6,
      mod7,
      mod8,
      mod9,
      mod10,
      mod11.0,
      mod11.1,
      test = "F")

## Analysis of Deviance Table
## 
## Model  1: mean_pot_count ~ s(distance, k = 5)
## Model  2: mean_pot_count ~ sum_rain + s(distance, k = 5)
## Model  3: mean_pot_count ~ mws + s(distance, k = 5)
## Model  4: mean_pot_count ~ sum_rain + mws + s(distance, k = 5)
## Model  5: mean_pot_count ~ sum_rain + s(distance + mws, k = 5)
## Model  6: mean_pot_count ~ sum_rain + s(distance, k = 5) + s(mws, k = 5)
## Model  7: mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5)
## Model  8: mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5) + s(sum_rain, 
##     k = 5)
## Model  9: mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5)
## Model 10: mean_pot_count ~ s(distance, k = 5) + s(sum_rain, k = 5) + mws
## Model 11: mean_pot_count ~ s(distance, k = 5) + s(mws, k = 5) + s(sum_rain, 
##     k = 5)
## Model 12: mean_pot_count ~ s(distance, k = 5, bs = "ts") + s(mws, k = 5, 
##     bs = "ts") + s(sum_rain, k = 5, bs = "ts")
##    Resid. Df Resid. Dev          Df Deviance          F    Pr(>F)    
## 1     329.00     248.10                                              
## 2     328.00     246.34  1.00004467     1.76     4.8398  0.028515 *  
## 3     328.00     236.49  0.00032438     9.84 83384.0223 2.780e-11 ***
## 4     327.00     233.31  1.00009343     3.19     8.7512  0.003321 ** 
## 5     328.00     246.34 -1.00041782   -13.03    35.7856 5.797e-09 ***
## 6     324.01     204.37  3.99780494    41.97    28.8451 < 2.2e-16 ***
## 7     325.01     206.98 -1.00194276    -2.62     7.1733  0.007744 ** 
## 8     324.01     204.38  0.99830140     2.60     7.1576  0.007873 ** 
## 9     326.81     230.54 -2.79549912   -26.16    25.7123 3.516e-14 ***
## 10    324.04     204.44  2.76976222    26.11    25.8972 3.572e-14 ***
## 11    323.66     639.52  0.37229260  -435.08                         
## 12    323.68     644.19 -0.01578706    -4.66   811.7846 2.141e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

0.3.4 Check Best Model Fit

0.3.4.1 Mod11.0 - s(Distance) + s(Windspeed) + s(Precipitation), family = tw()

mod11.0_vis <- getViz(mod11.0)
check(mod11.0_vis,
      a.qq = list(method = "tnorm", 
                  a.cipoly = list(fill = "light blue")), 
      a.respoi = list(size = 0.5), 
      a.hist = list(bins = 10))

## 
## Method: REML   Optimizer: outer newton
## full convergence after 8 iterations.
## Gradient range [-0.0006677089,-7.453262e-09]
## (score 309.7676 & scale 0.3639647).
## Hessian positive definite, eigenvalue range [0.3939723,2978.454].
## Model rank =  13 / 13 
## 
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
## 
##               k'  edf k-index p-value   
## s(distance) 4.00 3.50    0.86   0.010 **
## s(mws)      4.00 2.94    0.89   0.045 * 
## s(sum_rain) 4.00 1.90    0.90   0.045 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

0.3.4.2 Mod11.0 - s(Distance, family = “ts”) + s(Windspeed, family = “ts”) + s(Precipitation, family = “ts”), family = tw()

mod11.1_vis <- getViz(mod11.1)
check(mod11.1_vis,
      a.qq = list(method = "tnorm", 
                  a.cipoly = list(fill = "light blue")), 
      a.respoi = list(size = 0.5), 
      a.hist = list(bins = 10))

## 
## Method: REML   Optimizer: outer newton
## full convergence after 8 iterations.
## Gradient range [-6.815834e-09,5.749888e-08]
## (score 322.3273 & scale 0.3641249).
## Hessian positive definite, eigenvalue range [0.6649335,2979.663].
## Model rank =  13 / 13 
## 
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
## 
##               k'  edf k-index p-value   
## s(distance) 4.00 3.25    0.86   0.010 **
## s(mws)      4.00 2.88    0.88   0.020 * 
## s(sum_rain) 4.00 1.84    0.89   0.025 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

0.4 Thoughts

This model, mod11.0, mean_pot_count ~ s(Distance) + s(Windspeed) + s(Precipitation) - family = tw(), lacks enough numbers of basis functions for two of three predictors, distance, wsp, but it’s close. Realistically we need to increase the k values for these predictors, but we’re lacking the data to do this.